Let $\Phi_m(X)$ be the $m$-th cyclotomic polynomial, $p$ be a prime, and $\mathbb{Z}_p[X]/\Phi_m(X)$ be a quotient ring.
I believe that $\mathcal{Gal}(\mathbb{Q}(\zeta)\mathbb{Q}) \cong (\mathbb{Z}/m\mathbb{Z})^*$ but, I do not know how it works, where $\zeta$ is the primitive $m$-th root of unity ($\zeta \in Z_p$ and p > m).
What happens when applying the galois group permutation on an element over $\mathbb{Z}_p[X]/\Phi_m(X)$.
For example, if $m$=8 and $p$=17. Then the quotient ring is $\mathbb{Z}_{17}[X]/(X^4+1)$ and an element over the ring is $a=a_0 + a_1 X + a_2 X^2 + a_3 X^3 \in \mathbb{Z}_{17}[X]/(X^4+1)$, $a_i\in \mathbb{Z}_{17}$.
So, What is the corresponding operations of $\mathcal{Gal}(\mathbb{Q}(\zeta)/\mathbb{Q})$ and what's happening after applying that?